Periodic points of holomorphic maps via Lefschetz numbers

Fagella, Núria; Llibre, Jaume

doi:10.1090/S0002-9947-00-02608-8

Periodic points of holomorphic maps via Lefschetz numbers
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by Núria Fagella and Jaume Llibre PDF

Trans. Amer. Math. Soc. 352 (2000), 4711-4730 Request permission

Abstract:

In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension $n$ and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of ${\mathbb CP}(n)$ of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker’s theorem.

References

V. I. Arnold, Aspects des systèmes dynamiques, Preprint Ecole Polytechnique, Journées X-UPS 1994. Also in Topological Methods in Nonlinear Analysis, Torum, 1994.
I. K. Babenko and S. A. Bogatyĭ, Behavior of the index of periodic points under iterations of a mapping, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 1, 3–31 (Russian); English transl., Math. USSR-Izv. 38 (1992), no. 1, 1–26. MR 1130026, DOI 10.1070/IM1992v038n01ABEH002185
I. N. Baker, Fixpoints of polynomials and rational functions, J. London Math. Soc. 39 (1964), 615–622. MR 169989, DOI 10.1112/jlms/s1-39.1.615
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0283793
R. Grimshaw, Nonlinear ordinary differential equations, Applied Mathematics and Engineering Science Texts, CRC Press, Boca Raton, FL, 1993. MR 1311023
A. Dold, Fixed point indices of iterated maps, Invent. Math. 74 (1983), no. 3, 419–435. MR 724012, DOI 10.1007/BF01394243
John M. Franks, Homology and dynamical systems, CBMS Regional Conference Series in Mathematics, vol. 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C. by the American Mathematical Society, Providence, R.I., 1982. MR 669378, DOI 10.1090/cbms/049
S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479–494. MR 0000088, DOI 10.1007/BF03046993
John Erik Fornæss and Nessim Sibony, Complex dynamics in higher dimension. I, Astérisque 222 (1994), 5, 201–231. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). MR 1285389
John Guaschi and Jaume Llibre, Periodic points of $\scr C^1$ maps and the asymptotic Lefschetz number, Proceedings of the Conference “Thirty Years after Sharkovskiĭ’s Theorem: New Perspectives” (Murcia, 1994), 1995, pp. 1369–1373. MR 1361925, DOI 10.1142/S0218127495001046
A. Guillamon, X. Jarque, J. Llibre, J. Ortega, and J. Torregrosa, Periods for transversal maps via Lefschetz numbers for periodic points, Trans. Amer. Math. Soc. 347 (1995), no. 12, 4779–4806. MR 1321576, DOI 10.1090/S0002-9947-1995-1321576-9
Antoni A. Kosinski, Differential manifolds, Pure and Applied Mathematics, vol. 138, Academic Press, Inc., Boston, MA, 1993. MR 1190010
Solomon Lefschetz, Differential equations: Geometric theory, 2nd ed., Pure and Applied Mathematics, Vol. VI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0153903
Jaume Llibre, Lefschetz numbers for periodic points, Nielsen theory and dynamical systems (South Hadley, MA, 1992) Contemp. Math., vol. 152, Amer. Math. Soc., Providence, RI, 1993, pp. 215–227. MR 1243477, DOI 10.1090/conm/152/01325
Takashi Matsuoka, The number of periodic points of smooth maps, Ergodic Theory Dynam. Systems 9 (1989), no. 1, 153–163. MR 991495, DOI 10.1017/S0143385700004879
Ivan Niven and Herbert S. Zuckerman, An introduction to the theory of numbers, 4th ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 572268
Norbert Steinmetz, Rational iteration, De Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1993. Complex analytic dynamical systems. MR 1224235, DOI 10.1515/9783110889314
M. Shub and D. Sullivan, A remark on the Lefschetz fixed point formula for differentiable maps, Topology 13 (1974), 189–191. MR 350782, DOI 10.1016/0040-9383(74)90009-3
James W. Vick, Homology theory, 2nd ed., Graduate Texts in Mathematics, vol. 145, Springer-Verlag, New York, 1994. An introduction to algebraic topology. MR 1254439, DOI 10.1007/978-1-4612-0881-5
Joachim Wehler, Versal deformation of Hopf surfaces, J. Reine Angew. Math. 328 (1981), 22–32. MR 636192, DOI 10.1515/crll.1981.328.22
Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0387634